Question

How do you find the lengths of the arc of a circle of radius 9 feet intercepted by the central angle ${{60}^{\circ }}$?

Answer 1

We first find the perimeter of the circle of radius 9 feet. We know if we are trying to find the length of a partition of the perimeter with a known angle at the centre then we can find the length based on the proportionality. We find the length by that theorem.

Complete step by step solution:
The given circle is of radius 9 feet. We have to find the lengths of the arc of that circle intercepted by the central angle ${{60}^{\circ }}$.

The centre of the circle is O. $\angle BOC={{60}^{\circ }}$. $OB=OC=9$. We need to find the length of $\overset\frown{BC}$.
The central angle of the circle at the centre is ${{360}^{\circ }}$. This total angle is responsible for the whole perimeter of the circle.
Now if we are trying to find the length of a partition of the perimeter with a known angle at the centre then we can find the length based on the proportionality.
As the angle ${{360}^{\circ }}$ is attached for the whole perimeter then the angle ${{60}^{\circ }}$ is attached for the required arc length.
The perimeter of a circle with radius r unit is $2\pi r$ unit. For value of $r=9$ we have the perimeter as $2\pi r=18\pi$ unit.
The ratio of our required angle and the total angle is $\dfrac{{{60}^{\circ }}}{{{360}^{\circ }}}=\dfrac{1}{6}$.
Let’s assume the length of the arc is $a$ unit. Then the ratio is $\dfrac{a}{18\pi }$. This should be equal to $\dfrac{1}{6}$.
So, $\dfrac{a}{18\pi }=\dfrac{1}{6}\Rightarrow a=\dfrac{18\pi }{6}=3\pi$.
The approximate value is $3\pi =9.42$ feet. The arc of a circle of radius 9 feet intercepted by the central angle ${{60}^{\circ }}$ is $9.42$ feet.

Note: We need to remember that we can also use proportionality. The theorem is also applicable for the area of the circle. The area, perimeter and the central angle are all connected for a circle.